Not in my last 6 flights they haven't, at least not without trying to be incredibly covert about it which I seriously doubt. All these flights were within Europe or SE Asia, I don't know if head counts are more common in other regions.

Within the US they definitely count the passengers. I flew between Canada and Asia last year and I don't remember if they counted or not, but on flights within the USA they definitely do count. There was a rather embarrassing incident where a minor without a ticket of any kind got on a plane in the US and nobody ever did anything to make sure he was in the right place or even had a ticket for the flight. I think now all the airlines want to make sure that kind of thing never happens again, because if a kid can do it, an adult with bad intentions may be able to do ti too.

They don't compare the count to the number of tickets. They compare it to the number of people known to be getting on the flight, presumably these days from the number who've been scanned through security (in my airside days it was the number that had checked in at the desk, since this was before online check-in).

.. Which this device claims to be able to get through (the jetway is after the last ticket check). So the numbers may actually match up...